The harmonic index is a topological index based on the degrees of vertices in a graph. Introduced by Fajtlowicz in 1987, it has become an important descriptor in chemical graph theory and has been extensively studied for its mathematical properties and applications.
Definition For a simple connected graph G = ( V , E ) {\displaystyle G=(V,E)} with vertex set V {\displaystyle V} and edge set E {\displaystyle E} , the harmonic index H ( G ) {\displaystyle H(G)} is defined as:
H ( G ) = ∑ u v ∈ E ( G ) 2 d ( u ) + d ( v ) {\displaystyle \displaystyle H(G)=\sum _{uv\in E(G)}{\frac {2}{d(u)+d(v)}}}
where d ( u ) {\displaystyle d(u)} and d ( v ) {\displaystyle d(v)} denote the degrees of vertices u {\displaystyle u} and v {\displaystyle v} respectively, and the sum is taken over all edges u v {\displaystyle uv} in G {\displaystyle G} .
Relationship to other indices The harmonic index is closely related to the Randić index and can be considered as one of its variants. While the Randić index uses ( d ( u ) ⋅ d ( v ) ) − 1 / 2 {\displaystyle (d(u)\cdot d(v))^{-1/2}} as the edge weight, the harmonic index uses the harmonic mean of the endpoint degrees. The harmonic index is also related to the inverse degree of a graph:
I D ( G ) = ∑ u ∈ V ( G ) 1 d ( u ) {\displaystyle \displaystyle ID(G)=\sum _{u\in V(G)}{\frac {1}{d(u)}}}
Properties
Bounds For a connected graph G {\displaystyle G} with n {\displaystyle n} vertices, m {\displaystyle m} edges, maximum degree Δ {\displaystyle \Delta } , minimum degree δ {\displaystyle \delta } , and p {\displaystyle p} pendant edges: Lower bound:
H ( G ) ≥ 2 p Δ + 1 + m − p Δ {\displaystyle \displaystyle H(G)\geq {\frac {2p}{\Delta +1}}+{\frac {m-p}{\Delta }}}
Equality holds if and only if G {\displaystyle G} is the star graph K 1 , n − 1 {\displaystyle K_{1,n-1}} , a regular graph, or a ( Δ , 1 ) {\displaystyle (\Delta ,1)} -semiregular graph. Upper bounds: Various upper bounds have been established for specific graph classes. For example, for trees:
For any tree T {\displaystyle T} of order n ≥ 3 {\displaystyle n\geq 3} : H ( T ) ≤ H ( P n ) = n − 3 4 + 4 3 {\displaystyle H(T)\leq H(P_{n})={\frac {n-3}{4}}+{\frac {4}{3}}} , with equality if and only if T {\displaystyle T} is the path graph P n {\displaystyle P_{n}}
For any tree T {\displaystyle T} of order n ≥ 3 {\displaystyle n\geq 3} : H ( T ) ≥ H ( S n ) = 2 ( n − 1 ) n {\displaystyle H(T)\geq H(S_{n})={\frac {2(n-1)}{n}}} , with equality if and only if T {\displaystyle T} is the star graph S n {\displaystyle S_{n}}
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